If there is no Subscribe link in OneCare on the laptop, it has picked up the login you apparently provided when in trial mode to test additional functions that only work when you provide a LiveID. Since you then activated a subscription with that same LiveID on the other PC, those credentials (the subscription status) would be exchanged down to the OneCare install during its check (daily if expiration of the trial is imminent). Note that if you installed the trial without ever providing a LiveID after installation, the "automatic" activation could not have happened.
s-teve